Here is the number of new permanent residents to Canada, as a percentage of the existing population, over time:
"New permanent residents" is not a perfect measure of immigration - it excludes temporary foreign workers, who have become much more important in recent years, and also ignores emigration, that is, the non-trivial number of newcomers who return to their home country. But it is as good a measure as we can get.
This chart, together with some basic information about Canadian economic history, demonstrates why it is hard to establish any kind of relationship between immigration and growth.
Sometimes - as when Canada accepted more than 37,000 refugees from the Hungarian uprising - immigration levels are determined by external events. But more often, immigration policy is a response to a domestic political agenda. The lowest immigration levels in the post-war period were experienced during the late 70s and early 80s. These were the years when last, and biggest, cohorts of the baby boom were hitting the labour market, and Canada's economic performance left something to be desired. Cutting immigration was a politically sensible move. But when the economy started to rebound in the 1990s, so did immigration targets.
Unfortunately the immigration data does not go back to the 1930s, but a similar pattern holds for that period. Severe economic depression was met with calls to restrict immigration, and the governments of the day were happy to respond.
The level of immigration is, to a significant extent, driven by economic conditions. Consequently, it is extremely difficult to know whether or not immigration promotes economic growth If we see a positive relationship between immigration and growth in the data, we don't know if that relationship exists because immigration causes growth, or because growth causes immigration. (This paper argues growth drives immigration while this paper disagrees).
Some researchers have tried to solve this problem by looking at the impact of immigration at a sub-national level using city or regional data. This approach, however, is subject to the difficulty that immigrants tend to want to live there are jobs. If we see that cities with large immigrant populations grow more rapidly than those with few immigrants, is it because immigration drives growth and innovation, or because no sensible immigrant would want to move to a place with few job opportunities? Economists have used a variety of clever techniques to get around this "endogeneity" problem - an excellent though dated (1995) survey of the literature by Friedberg and Hunt and can be found here. Their conclusion:
The theoretical literature on immigration and economic growth suggests that the impact of' immigrants on natives' income growth depends crucially on the human capital levels of the immigrants. Empirical research on this question has yielded conflicting answers, and more work on this issue is needed.
In the two decades since Friedberg and Hunt wrote that passage, more work on this issue has been carried out. Much of the research on this topic has been carried out using US data. However immigrants to the US, unlike immigrants to Canada, typically have lower human capital than the native born. There is a large group of scholars doing excellent work Canadian immigration policy. Here are links to fairly readable pieces by Ana Ferrer, David Green, Ted MacDonald, Mikal Skuterud, Chris Worswick, and Casey Warman. Other notable names (at the risk of omitting people) are Arthur Sweetman, Garnett Picot, Charlie Beach and Canadian-in-exile David Card.
I am very doubtful that it is possible to predict, with any degree of certainty, how the current government's plan to "substantially increase" immigration levels will impact the Canadian economy - and the well-being of both native born and immigrant Canadians - in the long run. In the past, we might have feared that immigrant engineers would end up driving taxis. Now that uber and self-driving cars are wiping out taxi driver jobs, it's hard to know what will happen.